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Yuen, Manwai

doi:10.1016/j.cnsns.2014.06.027

Cited by 21 publications

(4 citation statements)

References 15 publications

(17 reference statements)

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“…For example, the spherically symmetric solutions and related exact solutions for Euler equations in R N were established in [12,18,22] and references therein. While, Zhang and Zheng in [26] constructed rotational solutions for the Euler equations in 2D space with γ = 2, which were later generalized by Yuen in [24] to the following form in 3D space:…”

Section: Introduction 1vacuum Free Boundary Problem and Related Resultsmentioning

confidence: 99%

“…Remark 1.7 For the solution (1.22) and (1.23) to the Euler equations, if we set β = 0, u z = η is a constant, which is similar to the case of Li in [13]. On the other hand, when β > 0, u z = η a β (t) r β is a function of r and t, it is different from the case of Yuen in [24] where u z is given by n ′ (t) n(t) x 3 , which has nothing to do with variable r (see (1.4)).…”

Section: Remark 16mentioning

confidence: 97%

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Analytical solutions and asymptotic profiles of vacuum free boundary for the compressible Euler equations with time-dependent damping

Li,

Jia,

Zhang

et al. 2023

Preprint

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We constructed a class of analytical, rotationaland self-similar solutions to the isentropic compressible Euler equations with time-dependent damping $\frac{\mu}{\left( 1+t\right) ^{\lambda}}%\rho\mathbf{u}$ ($\mu\geq0$, $\lambda\in\mathbb{R}$) and vacuum free boundary in cylindrical coordinates. These solutions capture the precise physical vacuum behavior that the sound speed is $C^{1/2}$-H\"older continuous across the boundary and are determined by the freeboundary $a(t)$, which is the solution of a second order nonlinear ordinarydifferential equation with parameters $\mu$ and $\lambda$ (see (1.12)) $). Furthermore, global well-posedness and asymptotic behavior of the free boundary $a(t)$ are presented in this paper. Precisely, we show that for $\lambda>1$ the freeboundary will grow linearly in time, and radial velocity $u^{r}$ and axial velocity $u^{z}$ are bounded, while the angular velocity $u^{\phi}$ and the radial derivatives of velocity components all tend to zero as $t\rightarrow+\infty$ (see Remark 1.4). However, if $\lambda\leq1$, the free boundary will grow sub-linearly in time. In particular, if $-1\leq \lambda\leq 1/(2\gamma-1)$, where $\gamma$ is the adiabatic exponent of polytropicgases, the free boundary will grow more slowly as $\lambda$ becomes smaller, and possesses a finite upper bound when $\lambda<-1$. Mathematics Subject Classifications (2020): 35R35, 76N10

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Section: Introduction 1vacuum Free Boundary Problem and Related Resultsmentioning

confidence: 99%

Section: Remark 16mentioning

confidence: 97%

Analytical solutions and asymptotic profiles of vacuum free boundary for the compressible Euler equations with time-dependent damping

Li,

Jia,

Zhang

et al. 2023

Preprint

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“…If we replace the pressure p p 0 2 ( ) r = -m r by p(ρ) = Aρ γ with A > 0 and γ 1 being two constants, then the system (1.1) becomes the classical compressible Euler equations for the polytropic fluids. There are many papers about the analytical solutions of the compressible Euler and Navier-Stokes equations in the literature, see [34][35][36][37][38][39][40][41][42][43] and the references therein, but the analytical solutions to the Euler equations for Chaplygin gas are seldom. To our knowledge, Huang and Wang [44] constructed some analytical solutions to the radially symmetric multidimensional Euler equations for Chaplygin gas and investigated the blowup phenomena of the solutions, furthermore they also studied the global-in-time existence and blowup phenomena of the one-dimensional solutions by the constructing method.…”

Section: Introductionmentioning

confidence: 99%

Analytical solutions to the Euler equations for Chaplygin gas

Jiang,

Dong

2024

Phys. Scr.

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In this paper, we study the analytical solutions to the Euler equations for Chaplygin gas. First, we construct two exact solutions for the one-dimensional system by using a self-similar ansatz. Second, we present some analytical solutions for the N-dimensional radially symmetric system. Third, we extend the above results to the two-phase flow case. The concentration and cavitation phenomena are observed from the constructed solutions.

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“…The self-similar Ansatz and related constructions have been effectively applied in a number of hydrodynamics systems (Barna and Mátyás 2013, Yuen 2015, Chen et al 2017, Gugat and Ulbrich 2017, Vishwakarma et al 2018, Animasaun and Pop 2017. The book of Campos (Barna 2017) covers more methods related to Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

Self-similar analysis of a viscous heated Oberbeck–Boussinesq flow system

Barna¹,

Mátyás²,

Pocsai³

2020

Fluid Dyn. Res.

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The simplest model to couple the heat conduction and Navier-Stokes equations together is the Oberbeck-Boussinesq(OB) system which were investigated by E.N. Lorenz and opened the paradigm of chaos. In our former studies -Chaos Solitons and Fractals 78, 249 (2015), ibid, 103, 336 (2017) -we derived analytic solutions for the velocity, pressure and temperature fields. Additionally, we gave a possible explanation of the Rayleigh-Bènard convection cells with the help of the self-similar Ansatz. Now we generalize the OB hydrodynamical system, including a viscous source term in the heat conduction equation. Our results may attract the interest of various fields like micro or nanofluidics or climate studies.

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Rotational and self-similar solutions for the compressible Euler equations in R3

Abstract: In this paper, we present rotational and self-similar solutions for the compressible Euler equations in R 3 using the separation method.

Cited by 21 publications

References 15 publications

Analytical solutions and asymptotic profiles of vacuum free boundary for the compressible Euler equations with time-dependent damping

Analytical solutions and asymptotic profiles of vacuum free boundary for the compressible Euler equations with time-dependent damping

Analytical solutions to the Euler equations for Chaplygin gas

Self-similar analysis of a viscous heated Oberbeck–Boussinesq flow system

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